Optimal. Leaf size=257 \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}+\frac{7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
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Rubi [A] time = 0.332935, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {742, 640, 612, 621, 206} \[ \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac{\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}+\frac{7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\int \left (\frac{1}{2} \left (12 c d^2-2 e \left (\frac{5 b d}{2}+a e\right )\right )+\frac{7}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (-\frac{7}{2} b e (2 c d-b e)+c \left (12 c d^2-2 e \left (\frac{5 b d}{2}+a e\right )\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac{\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}-\frac{\left (\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{128 c^3}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{512 c^4}\\ &=-\frac{\left (b^2-4 a c\right ) \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{512 c^4}+\frac{\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac{7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac{\left (b^2-4 a c\right )^2 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.338076, size = 188, normalized size = 0.73 \[ \frac{\frac{\left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{512 c^{7/2}}+\frac{7 e (a+x (b+c x))^{5/2} (2 c d-b e)}{10 c}+e (d+e x) (a+x (b+c x))^{5/2}}{6 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 922, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.05072, size = 2044, normalized size = 7.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13443, size = 625, normalized size = 2.43 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x e^{2} + \frac{24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac{120 \, c^{6} d^{2} + 264 \, b c^{5} d e + 3 \, b^{2} c^{4} e^{2} + 140 \, a c^{5} e^{2}}{c^{5}}\right )} x + \frac{360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e + 768 \, a c^{5} d e - 7 \, b^{3} c^{3} e^{2} + 36 \, a b c^{4} e^{2}}{c^{5}}\right )} x + \frac{120 \, b^{2} c^{4} d^{2} + 2400 \, a c^{5} d^{2} - 120 \, b^{3} c^{3} d e + 672 \, a b c^{4} d e + 35 \, b^{4} c^{2} e^{2} - 216 \, a b^{2} c^{3} e^{2} + 240 \, a^{2} c^{4} e^{2}}{c^{5}}\right )} x - \frac{360 \, b^{3} c^{3} d^{2} - 2400 \, a b c^{4} d^{2} - 360 \, b^{4} c^{2} d e + 2400 \, a b^{2} c^{3} d e - 3072 \, a^{2} c^{4} d e + 105 \, b^{5} c e^{2} - 760 \, a b^{3} c^{2} e^{2} + 1296 \, a^{2} b c^{3} e^{2}}{c^{5}}\right )} - \frac{{\left (24 \, b^{4} c^{2} d^{2} - 192 \, a b^{2} c^{3} d^{2} + 384 \, a^{2} c^{4} d^{2} - 24 \, b^{5} c d e + 192 \, a b^{3} c^{2} d e - 384 \, a^{2} b c^{3} d e + 7 \, b^{6} e^{2} - 60 \, a b^{4} c e^{2} + 144 \, a^{2} b^{2} c^{2} e^{2} - 64 \, a^{3} c^{3} e^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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